Blow-up dynamics for radial self-dual Chern-Simons-Schr\"odinger equation with prescribed asymptotic profile

Abstract

We construct finite energy blow-up solutions for the radial self-dual Chern-Simons-Schr\"odinger equation with a continuum of blow-up rates. Our result stands in stark contrast to the rigidity of blow-up of $H^{3}$ solutions proved by the first author for equivariant index $m \geq 1$, where the soliton-radiation interaction is too weak to admit the present blow-up scenarios. It is optimal (up to an endpoint) in terms of the range of blow-up rates and the regularity of the asymptotic profiles in view of the authors' previous proof of $H^{1}$ soliton resolution for the self-dual Chern-Simons-Schr\"odinger equation in any equivariance class. Our approach is a backward construction combined with modulation analysis, starting from prescribed asymptotic profiles and deriving the corresponding blow-up rates from their strong interaction with the soliton. In particular, our work may be seen as an adaptation of the method of Jendrej-Lawrie-Rodriguez (developed for energy critical equivariant wave maps) to the Schr\"odinger case. However, the Schr\"odinger nature of the equation (in particular, the lack of finite speed of propagation) and the optimal range (up to the $H^{1}$-endpoint) of our blow-up construction give rise to new challenges. Notably, the construction of (approximate) radiation from the prescribed asymptotic profile is one of our key novelties and might be of independent interest.